TLDR: This research paper presents the first finite-time convergence analysis for actor-critic reinforcement learning algorithms operating with reward functions that change over time. It demonstrates that these algorithms can achieve an O(1/√T) convergence rate, matching the best-known rates for static reward scenarios, provided the reward parameters evolve sufficiently slowly. The study specifically validates gradient-based reward updates, common in practical RL, as satisfying this condition. Additionally, the paper introduces an improved analysis of distribution mismatch under Markovian sampling, enhancing theoretical understanding even in static reward settings.
Reinforcement Learning (RL) has achieved remarkable success in various real-world applications, from game playing to robotics. At its core, RL involves an agent learning to make decisions by interacting with an environment and receiving rewards. Traditionally, theoretical analyses of RL algorithms assume that the reward function—the feedback mechanism guiding the agent—remains constant throughout the learning process. However, many practical RL techniques deviate from this assumption, employing reward functions that change or “evolve” over time.
These evolving reward functions are not arbitrary; they are intentionally designed to improve learning efficiency and performance. Common examples include:
Reward Shaping
This involves adding auxiliary rewards to the environment to guide the agent towards desired behaviors. These extra rewards can come from prior knowledge or be learned during training.
Entropy or KL Regularization
These techniques introduce a term into the learning objective that encourages the agent to explore more or to stay close to a reference policy. This effectively modifies the reward function based on the agent’s current policy or other factors.
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Curriculum Learning
Here, the agent starts by learning easier versions of a task with simplified rewards, and gradually progresses to more complex tasks as its skills improve. This naturally leads to a changing reward landscape.
While these methods are widely used and empirically successful, a crucial theoretical question has remained largely unanswered: how quickly can the reward function change while still ensuring that an RL algorithm converges and learns effectively? This paper, “Finite-time Convergence Analysis of Actor-Critic with Evolving Reward,” addresses this fundamental question by providing the first finite-time convergence analysis for a popular class of RL algorithms known as actor-critic methods, specifically when the reward function is evolving. You can read the full paper here: Finite-time Convergence Analysis of Actor-Critic with Evolving Reward.
Actor-critic algorithms are a cornerstone of modern RL, combining two components: an “actor” that learns the policy (how to act) and a “critic” that estimates the value of states or actions. The critic helps the actor improve its policy by providing feedback on the quality of its actions. The research focuses on a single-timescale actor-critic algorithm, meaning both the actor and critic update their parameters simultaneously using the same stream of experience, under Markovian sampling (where states are sampled sequentially from the environment).
The paper’s main finding is significant: it shows that even with an evolving reward function, the actor-critic algorithm can achieve an O(1/√T) convergence rate. This rate is considered the best-known for actor-critic methods with static rewards, meaning the algorithm’s efficiency is largely preserved. However, this holds true under a critical condition: the reward parameters must evolve “slowly enough.”
What does “slowly enough” mean in practice? The research provides a clear answer. If the reward function is updated using a gradient-based rule—a common approach in many practical algorithms that learn intrinsic rewards or adapt regularization strengths—and if these updates occur on the same timescale as the actor and critic updates, then the O(1/√T) convergence rate is maintained. This provides a strong theoretical backing for a wide array of existing and future RL techniques that rely on dynamic reward structures.
As a secondary, but equally important, contribution, the paper introduces a novel method for analyzing the “distribution mismatch” that arises from Markovian sampling. This refers to the difference between the actual distribution of states encountered by the agent and the ideal stationary distribution. This new analysis improves upon previous theoretical bounds for static-reward scenarios by a factor of log2 T, offering a more precise understanding of how sampling affects learning.
In essence, this work demonstrates that single-timescale actor-critic algorithms are surprisingly robust to changes in the reward function. It bridges a critical gap between theoretical understanding and empirical practice in reinforcement learning, offering a solid foundation for designing more effective and stable algorithms in dynamic environments. The findings pave the way for future research into more complex scenarios, such as those involving non-linear function approximation with neural networks, and the development of even more sophisticated reward-shaping techniques.
